1906. This shows that cq(n) is always an integer. Compare it with the formula von Sterneck It is easily shown from the definition that cq(n) is multiplicative when considered as a function of q for a fixed value of n: i.e. From the definition (or Kluyver's formula) it is straightforward to prove that, if p is a prime number, and if pk is a prime power where k > 1, This result and the multiplicative property can be used to prove This is called von Sterneck's arithmetic function. The equivalence of it and Ramanujan's sum is due to Hölder. Other properties of cq(n) For all positive integers q, For a fixed value of q the absolute value of the sequence is bounded by φ(q), and for a fixed value of n the absolute value of the sequence is bounded by n. If q > 1 Let m1, m2 > 0, m = lcm(m1, m2). Then Ramanujan's sums satisfy an orthogonality property: Let n, k > 0. Then known as the Brauer - Rademacher identity. If n > 0 and a is any integer, we also have due to Cohen. Table Ramanujan expansions If f(n) is an arithmetic function (i.e. a complex-valued function of the integers or natural numbers), then a convergent infinite series of the form: or of the form: where the , is called a Ramanujan expansion of f(n). Ramanujan found expansions of some of the well-known functions of number theory. All of these results are proved in an "elementary" manner (i.e. only using formal manipulations of series and the simplest results about convergence). The expansion of the zero function depends on a result from the analytic theory of prime numbers, namely that the series converges to 0, and the results for r(n) and r′(n) depend on theorems in an earlier paper. All the formulas in this section are from Ramanujan's 1918 paper. Generating functions The generating functions of the Ramanujan sums are Dirichlet series: is a generating function for the sequence cq(1), cq(2), ... where q is kept constant, and is a generating function for the sequence c1(n), c2(n), ... where n is kept constant. There is also the double Dirichlet series σk(n) σk(n) is the divisor function (i.e. the sum of the k-th powers of the divisors of n, including 1 and n). σ0(n), the number of divisors of n, is usually written d(n) and σ1(n), the